n_13488 - How to judge a series is convergent or divergent...

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How to judge a series is convergent or divergent 1. Definition Given a series n =1 a n = a 1 + a 2 + a 3 + · · · , let s n denote its nth partial sum: s n = n X i =1 a i = a 1 + a 2 + · · · + a n If the sequence s n is convergent and lim n →∞ s n = s exists as a real number, then the series a n is called convergent and we write a 1 + a 2 + · · · + a n + · · · = s or X n =1 a n = s The number s is called the sum of the series. Otherwise, the series is called divergent . 2. The Test for Divergence If lim n →∞ a n does not exist or if lim n →∞ a n 6 = 0, then the series n =1 a n is divergent. 3. The Integral Test Suppose f is a continuous, positive, decreasing function on [1 , ) and let a n = f ( n ). Then the series n =1 a n is convergent if and only if the improper integral R 1 f ( x ) dx is convergent. 4. The Comparision Tests 4.1 The Comarision Test Suppose that a n and b n are series with positive terms. (i) If b n is convergent and a n b n for all n , then a n is also convergent. (ii) If b n is divergent and a n b n for all n , then a n is also divergent. 4.2 The Limit Comparision Test Suppose that a n and b n are series with positive terms. If lim n →∞ a n b n = c where
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